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'''神经雪崩''' is a cascade of [[bursting|bursts]] of activity in [[Neurons|neuronal]] networks whose size distribution can be approximated by a [[power law]], as in [[critical sandpile models]] (Bak et al. 1987). Neuronal avalanches are seen in cultured and acute [[cortical slices]] (Beggs and Plenz, 2003; 2004). Activity in these slices of [[neocortex]] is characterized by brief bursts lasting tens of milliseconds, separated by periods of quiescence lasting several seconds. When observed with a [[multielectrode array]], the number of electrodes driven over threshold during a burst is distributed approximately like a power law. Although this phenomenon is highly robust and reproducible, its relation to physiological processes in the intact brain is currently not known.   

'''神经雪崩''' is a cascade of [[bursting|bursts]] of activity in [[Neurons|neuronal]] networks whose size distribution can be approximated by a [[power law]], as in [[critical sandpile models]] (Bak et al. 1987). Neuronal avalanches are seen in cultured and acute [[cortical slices]] (Beggs and Plenz, 2003; 2004). Activity in these slices of [[neocortex]] is characterized by brief bursts lasting tens of milliseconds, separated by periods of quiescence lasting several seconds. When observed with a [[multielectrode array]], the number of electrodes driven over threshold during a burst is distributed approximately like a power law. Although this phenomenon is highly robust and reproducible, its relation to physiological processes in the intact brain is currently not known.   

==实验观察==

==实验观察 Experimental Observations==

[[Image:雪崩示例.jpg|thumb|400px|right|Example of an avalanche. Seven frames are shown, where each frame represents activity on the electrode array during one 4 ms time step. An avalanche is a series of consecutively active frames that is preceded by and terminated by blank frames. Avalanche size is given by the total number of active electrodes. The avalanche shown here has a size of 9.]]

[[Image:雪崩示例.jpg|thumb|400px|right|Example of an avalanche. Seven frames are shown, where each frame represents activity on the electrode array during one 4 ms time step. An avalanche is a series of consecutively active frames that is preceded by and terminated by blank frames. Avalanche size is given by the total number of active electrodes. The avalanche shown here has a size of 9.]]

===幂律尺寸分布===

===幂律尺寸分布 Power law size distribution===

The movie illustrates that multi-channel data can be broken down into frames where there is no activity and where there is at least one active electrode, which may pick up the activity from several neurons. A sequence of consecutively active frames, bracketed by inactive frames, can be called an avalanche.  

The movie illustrates that multi-channel data can be broken down into frames where there is no activity and where there is at least one active electrode, which may pick up the activity from several neurons. A sequence of consecutively active frames, bracketed by inactive frames, can be called an avalanche.  

The example avalanche shown has a size of 9 because this is the total number of electrodes that were driven over threshold. Avalanche sizes are distributed in a manner that is nearly fit by a [[power law]]. Due to the limited number of electrodes in the array, the power law begins to bend downward in a cutoff well before the array size of 60. But for larger electrode arrays, the power law is seen to extend much further.  

The example avalanche shown has a size of 9 because this is the total number of electrodes that were driven over threshold. Avalanche sizes are distributed in a manner that is nearly fit by a [[power law]]. Due to the limited number of electrodes in the array, the power law begins to bend downward in a cutoff well before the array size of 60. But for larger electrode arrays, the power law is seen to extend much further.  

where <math>P(S)</math> is the probability of observing an avalanche of size <math>S\ ,</math> <math>\alpha</math> is the exponent that gives the slope of the power law in a log-log graph, and <math>k</math> is a proportionality constant. For experiments with [[slice culture]]s, the size distribution of avalanches of [[local field potential]]s has an exponent <math>\alpha\approx 1.5\ ,</math> but in recordings of spikes from a different array the exponent is <math>\alpha\approx2.1\ .</math> The reasons behind this difference in exponents are still being explored. It is important to note that a power law distribution is not what would be expected if activity at each electrode were driven independently. An ensemble of uncoupled, Poisson-like processes would lead to an exponential distribution of event sizes. Further, while power laws have been reported for many years in neuroscience in the temporal correlations of single time-series data (e.g., the power spectrum from [[Electroencephalogram|EEG]] (Linkenkaer-Hansen et al, 2001; Worrell et al, 2002), [[Fano factor|Fano]] or [[Allan factor]]s in [[Spike Statistics|spike count statistics]] (Teich et al, 1997), [[neurotransmitter]] secretion times (Lowen et al, 1997), [[ion channel]] fluctuations (Toib et al, 1998), interburst intervals in neuronal cultures (Segev et al, 2002)), they had not been observed from interactions seen in multielectrode data. Thus neuronal avalanches emerge from collective processes in a distributed network.

where <math>P(S)</math> is the probability of observing an avalanche of size <math>S\ ,</math> <math>\alpha</math> is the exponent that gives the slope of the power law in a log-log graph, and <math>k</math> is a proportionality constant. For experiments with [[slice culture]]s, the size distribution of avalanches of [[local field potential]]s has an exponent <math>\alpha\approx 1.5\ ,</math> but in recordings of spikes from a different array the exponent is <math>\alpha\approx2.1\ .</math> The reasons behind this difference in exponents are still being explored. It is important to note that a power law distribution is not what would be expected if activity at each electrode were driven independently. An ensemble of uncoupled, Poisson-like processes would lead to an exponential distribution of event sizes. Further, while power laws have been reported for many years in neuroscience in the temporal correlations of single time-series data (e.g., the power spectrum from [[Electroencephalogram|EEG]] (Linkenkaer-Hansen et al, 2001; Worrell et al, 2002), [[Fano factor|Fano]] or [[Allan factor]]s in [[Spike Statistics|spike count statistics]] (Teich et al, 1997), [[neurotransmitter]] secretion times (Lowen et al, 1997), [[ion channel]] fluctuations (Toib et al, 1998), interburst intervals in neuronal cultures (Segev et al, 2002)), they had not been observed from interactions seen in multielectrode data. Thus neuronal avalanches emerge from collective processes in a distributed network.

===重复的雪崩模式===

===重复的雪崩模式 Repeating avalanche patterns===

[[Image:急性切片的重复雪崩的家族.jpg|thumb|200px|left|Families of repeating avalanches from an acute slice. Each family (1-4) shows a group of three similar avalanches.  Similarity within each group was higher than expected by chance when compared to 50 sets of shuffled data. Repeating avalanches also occur in cortical [[slice culture]]s, where there are on average 30 ± 14 (mean ± s.d.) distinct families of reproducible avalanches, each containing about 23 avalanches (Beggs and Plenz, 2004). Repeating avalanches are stable for 10 hrs and have a temporal precision of 4 ms, suggesting that they could serve as a substrate for storing information in [[neural networks]].]]  

[[Image:急性切片的重复雪崩的家族.jpg|thumb|200px|left|Families of repeating avalanches from an acute slice. Each family (1-4) shows a group of three similar avalanches.  Similarity within each group was higher than expected by chance when compared to 50 sets of shuffled data. Repeating avalanches also occur in cortical [[slice culture]]s, where there are on average 30 ± 14 (mean ± s.d.) distinct families of reproducible avalanches, each containing about 23 avalanches (Beggs and Plenz, 2004). Repeating avalanches are stable for 10 hrs and have a temporal precision of 4 ms, suggesting that they could serve as a substrate for storing information in [[neural networks]].]]  

Power law distributions of sequence sizes have also been observed in spikes from the isolated [[leech ganglion]] (V. Torre, conference talk) and in spikes from [[dissociated cortical cultures]] (L. Bettencourt; R. Alessio,  personal communications), suggesting that the phenomenon of avalanches may be quite general to in-vitro preparations. Preliminary reports also indicate that avalanches are present in the superficial cortical layers of awake, resting primates (Petermann et al, 2006). These reports have not been published yet and are included here only to indicate that researchers are now exploring the avalanche concept in a variety of preparations.

Power law distributions of sequence sizes have also been observed in spikes from the isolated [[leech ganglion]] (V. Torre, conference talk) and in spikes from [[dissociated cortical cultures]] (L. Bettencourt; R. Alessio,  personal communications), suggesting that the phenomenon of avalanches may be quite general to in-vitro preparations. Preliminary reports also indicate that avalanches are present in the superficial cortical layers of awake, resting primates (Petermann et al, 2006). These reports have not been published yet and are included here only to indicate that researchers are now exploring the avalanche concept in a variety of preparations.

==雪崩模型==  

==雪崩模型 Models of avalanches==  

[[Image:分支过程的三个阶段.jpg|thumb|200px|right|The three regimes of a branching process. Top, when the branching parameter, <math>\sigma\ ,</math> is less than unity, the system is subcritical and activity dies out over time. Middle, when the branching parameter is equal to unity, the system is critical and activity is approximately sustained. In actuality, activity will die out very slowly with a power law tail. Bottom, when the branching parameter is greater than unity, the system is supercritical and activity increases over time.]]

[[Image:分支过程的三个阶段.jpg|thumb|200px|right|The three regimes of a branching process. Top, when the branching parameter, <math>\sigma\ ,</math> is less than unity, the system is subcritical and activity dies out over time. Middle, when the branching parameter is equal to unity, the system is critical and activity is approximately sustained. In actuality, activity will die out very slowly with a power law tail. Bottom, when the branching parameter is greater than unity, the system is supercritical and activity increases over time.]]

where <math>\sigma_i</math> is the expected number of descendant processing units activated by unit <math>i\ ,</math> <math>N</math> is the number of units that unit <math>i</math> connects to, and <math>p_{ij}</math> is the probability that activity in unit <math>i</math> will transmit to unit <math>j\ .</math> Because some transmission probabilities are greater than others, preferred paths of transmission may occur, leading to reproducible avalanche patterns. Both the power law distribution of avalanche sizes and the repeating avalanches are qualitatively captured by this model when <math>\sigma</math> is tuned to the critical point (<math>\sigma=1</math>), as shown in the figure (Haldeman and Beggs, 2005). When the model is tuned moderately above (<math>\sigma>1</math>) or below (<math>\sigma<1</math>) the critical point, it fails to produce a power law distribution of avalanche sizes. This phenomenological model does not explicitly state the cellular or synaptic mechanisms that may underlie the branching process, and many of this model's predictions need to be tested.

where <math>\sigma_i</math> is the expected number of descendant processing units activated by unit <math>i\ ,</math> <math>N</math> is the number of units that unit <math>i</math> connects to, and <math>p_{ij}</math> is the probability that activity in unit <math>i</math> will transmit to unit <math>j\ .</math> Because some transmission probabilities are greater than others, preferred paths of transmission may occur, leading to reproducible avalanche patterns. Both the power law distribution of avalanche sizes and the repeating avalanches are qualitatively captured by this model when <math>\sigma</math> is tuned to the critical point (<math>\sigma=1</math>), as shown in the figure (Haldeman and Beggs, 2005). When the model is tuned moderately above (<math>\sigma>1</math>) or below (<math>\sigma<1</math>) the critical point, it fails to produce a power law distribution of avalanche sizes. This phenomenological model does not explicitly state the cellular or synaptic mechanisms that may underlie the branching process, and many of this model's predictions need to be tested.

==Implications of avalanches==

==雪崩的含义 Implications of avalanches==

When a tunable system operates in a regime where it produces power law distributions, it is said to be operating at the [[Self-Organized Criticality|critical point]]. Strictly speaking, only infinitely large systems can operate at the critical point, but here the term “critical” is used to describe behavior in finite systems that would approach criticality if they were extended to unlimited sizes. The power law avalanche size distribution has potential implications for information processing in neural networks in these four areas:

When a tunable system operates in a regime where it produces power law distributions, it is said to be operating at the [[Self-Organized Criticality|critical point]]. Strictly speaking, only infinitely large systems can operate at the critical point, but here the term “critical” is used to describe behavior in finite systems that would approach criticality if they were extended to unlimited sizes. The power law avalanche size distribution has potential implications for information processing in neural networks in these four areas:

Optimizing all of these information processing tasks may occur simultaneously when a network operates near the critical point, where neuronal avalanches occur.  

Optimizing all of these information processing tasks may occur simultaneously when a network operates near the critical point, where neuronal avalanches occur.  

==Relationship of neuronal avalanches to other systems==

==神经雪崩与其他系统的关系 Relationship of neuronal avalanches to other systems==

Power law distributions of event sizes are often seen in complex phenomena including earthquakes, [[phase transitions]], [[percolation]], forest fires, financial market fluctuations, avalanches in the [[game of life]] and a host of others (Bak, 1996). In some specific cases, this similarity appears to be more than superficial. For example, earthquake models incorporate local rules in which forces at one site are distributed to nearest neighbors without dissipation. This conservation of forces is similar to the conservation of probabilities in the critical branching model described above. This suggests that conservation of synaptic strengths, as reported in (Royer and Pare, 2003) could be a mechanism responsible for maintaining a network near the critical point. In a related idea, simulations indicate that networks can be kept nearly critical when the total sum of synaptic strengths hovers near a constant value (Hsu and Beggs, 2006). This could be accomplished through a mechanism like synaptic scaling (Turrigiano and Nelson, 2000), which has been observed experimentally. Finally, recently "burned" areas in forest fire models are refractory, while unburned areas are more likely to ignite. This balance of refractoriness and excitability combine to maintain the system near the critical point. Recent models of neuronal avalanches (Levina, Herrmann and Geisel, 2005) have suggested that short-term synaptic depression and facilitation may also serve to drive neuronal networks toward the critical point where avalanches occur. Thus, an understanding of power laws in diverse complex systems can suggest mechanisms that might underlie criticality in neuronal networks.   

Power law distributions of event sizes are often seen in complex phenomena including earthquakes, [[phase transitions]], [[percolation]], forest fires, financial market fluctuations, avalanches in the [[game of life]] and a host of others (Bak, 1996). In some specific cases, this similarity appears to be more than superficial. For example, earthquake models incorporate local rules in which forces at one site are distributed to nearest neighbors without dissipation. This conservation of forces is similar to the conservation of probabilities in the critical branching model described above. This suggests that conservation of synaptic strengths, as reported in (Royer and Pare, 2003) could be a mechanism responsible for maintaining a network near the critical point. In a related idea, simulations indicate that networks can be kept nearly critical when the total sum of synaptic strengths hovers near a constant value (Hsu and Beggs, 2006). This could be accomplished through a mechanism like synaptic scaling (Turrigiano and Nelson, 2000), which has been observed experimentally. Finally, recently "burned" areas in forest fire models are refractory, while unburned areas are more likely to ignite. This balance of refractoriness and excitability combine to maintain the system near the critical point. Recent models of neuronal avalanches (Levina, Herrmann and Geisel, 2005) have suggested that short-term synaptic depression and facilitation may also serve to drive neuronal networks toward the critical point where avalanches occur. Thus, an understanding of power laws in diverse complex systems can suggest mechanisms that might underlie criticality in neuronal networks.